Question

The difference between S.I. and C.I. compounded annually on a certain sum of money for $ 2\,years $ at $ 8\% $ per annum is ₹. $ 12.80 $ . Find the principal.

Answer 1

Hint: Here, we first let principal be as ‘x’ then calculating simple interest (S.I.) and compound interest (C.I.) by suing formula of simple interest and compound interest and equating their difference equal to amount given in problem to form an equation. While solving we can find the value of ‘x’, hence required value of principal.
Formula of simple interest (S.I.) = $ \dfrac{{P \times R \times T}}{{100}} $ , Formula for amount in case of compound interest: $ A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T} $ , Compound Interest (C.I.) = A – P.

Complete step-by-step answer:
First let value of principal be = x
Time for which amount is taken = $ 2\,years $
Rate percent per annum = $ 8\% $
Simple interest on amount ‘x’ is given as:
 $ Simple\,\,Interest(S.I.) = \dfrac{{P \times R \times T}}{{100}} $ ,
Where P is the principal, R is rate percent and T is time for the period amount is taken as a loan.
Substituting values in above formula we have:
 $
  Simple\,\,Interest(S.I.) = \dfrac{{x\, \times \,8 \times \,2}}{{100}} \\
   = \dfrac{{4x}}{{25}} \;
  $
Hence, from above we see that principal on ₹. (x) for $ 2\,years $ at $ 8\% $ per annum is ₹. $ \dfrac{{4x}}{{25}} $ .
To find compound interest on principal ‘x’ we first calculate the amount and then on find the difference of amount and principal to get compound interest.
 $ Amount = P{\left( {1 + \dfrac{R}{{100}}} \right)^T} $
Substituting values in above formula we have:
 $
  Amount = x{\left( {1 + \dfrac{8}{{100}}} \right)^2} \\
   \Rightarrow Amount(A) = x{\left( {1 + \dfrac{2}{{25}}} \right)^2} \\
   \Rightarrow Amount(A) = x{\left( {\dfrac{{27}}{{25}}} \right)^2} \\
   \Rightarrow Amount(A) = \dfrac{{729}}{{625}}x \;
  $
Now, to find compound interest for this we calculate the difference of the amount obtained and principal given.
 $
  Compound\,\,Interest(C.I.) = Amount(A) - Principal(P) \\
   \Rightarrow Compound\,\,Interest(C.I.) = \dfrac{{729}}{{625}}x - x \\
   \Rightarrow Compound\,\,Interest(C.I.) = \dfrac{{729x - 625x}}{{625}} \\
   \Rightarrow Compound\,\,Interest(C.I.) = \dfrac{{104}}{{625}}x \;
  $
From above we see that compound interest on principal ‘x’ is ₹. $ \dfrac{{104}}{{625}}x $ .
Now, finding the difference of compound interest and simple interest calculated above and equated to the difference of the amount given in question.
 $ \Rightarrow C.I. - S.I. = 12.80 $
Therefore, from above we have:
 $
  \dfrac{{104}}{{625}}x - \dfrac{4}{{25}}x = 12.80 \\
   \Rightarrow \dfrac{{104x - 100x}}{{625}} = 12.80 \\
   \Rightarrow \dfrac{{4x}}{{625}} = 12.80 \\
   \Rightarrow x = 12.80 \times \dfrac{{625}}{4} \;
   \Rightarrow x = 2000 \;
  $
Hence, from above we see that principal is ₹. $ 2000 $

Note: Compound interest on any amount can be calculated in two ways. In the first way one can first find the amount of principal and then calculate the difference of the amount obtained and principal given to find compound interest. In second way one can directly apply formula $ P\left\{ {{{\left( {1 + \dfrac{R}{{100}}} \right)}^T} - 1} \right\} $ to find compound interest on given principal.